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Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential
Relationships; Topic Notes
STRAND 3: QUADRATIC AND NONLINEAR FUNCTIONS
TOPIC 3.3: EXPONENTIAL RELATIONSHIPS
Topic Notes
Mathematical focus
Participants explore exponential growth and decay situations. Participants
develop the ideas of the common multiplier or ratio as the base of an exponential
function and the starting point as the y-intercept of an exponential function. The
development of exponential functions begins with investigating repeated addition
and repeated multiplication using the graphing calculator. Then, exponential
growth and exponential decay are investigated. Exponential growth and decay are
identified as a result of repeated multiplication.
Terms: growth, decay, repeated addition, repeated multiplication.
Topic overview
This topic contains 4 tasks:
Task 3.3.1: Calculator Model for Exponential Functions
Task 3.3.2: One Grain of Rice
Task 3.3.3: Stars, Stars, Stars
Task 3.3.4: Exponential Decay: No Beans About it
In task 3.3.1, participants investigate repeated addition and repeated multiplication
using the graphing calculator. Participants use the sequence of functions to
repeatedly add and multiply on the calculator. They recognize and interpret
repeated addition and repeated multiplication symbolically, through tables, and
graphs.
In task 3.3.2, participants use the book by Demi, One Grain of Rice, to develop the
concept of exponential growth.
In task 3.3.3, participants use the context of placing stars on a sheet of paper,
separated into 16 sections, to develop the concept of exponential growth. They
investigate the effect of each of these symbolically, through tables, and graphs.
In task 3.3.4, participants use the context of removing one-third of the beans from a
container to investigate exponential decay. They investigate the effect of each of
these symbolically through tables, and graphs.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
2
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential
Relationships; Topic Notes
TExES Standards focus
TExES Standard II.004 Patterns and algebra. The teacher uses patterns to
model and solve problems and formulate conjectures. The beginning
teacher:
(B)
Uses methods of recursion and iteration to model and solve
problems.
TExES Standard II.005 Patterns and algebra. The teacher understands
attributes of functions, relations, and their graphs. The beginning teacher:
(B)
Identifies the mathematical domain and range of functions and
relations and determines reasonable domains for given situations.
(C)
Understands that a function represents a dependence of one
quantity on another and can be represented in a variety of ways
(e.g., concrete models, tables, graphs, diagrams, verbal
descriptions, symbols).
TExES Standard II.008 Patterns and algebra. The teacher understands
exponential and logarithmic functions, analyzes their algebraic and
graphical properties, and uses them to model and solve problems. The
beginning teacher:
(A)
Recognizes and translates among various representations (e.g.,
written, numerical, tabular, graphical, algebraic) of exponential
and logarithmic functions.
TEKS/TAKS focus
TEKS A.3 Foundations of functions. The student understands how algebra can
be used to express generalizations and recognizes and uses the power of
symbols to represent situations. The student is expected to:
(B)
look for patterns and represent generalizations algebraically.
High School TAKS Objective 2: The student will demonstrate an understanding
of the properties and attributes of functions.
TEKS A.11 Quadratic and other nonlinear functions. The student understands
there are situations modeled by functions that are neither linear nor
quadratic and models the situations. The student is expected to:
(C)
analyze data and represent situations involving exponential growth
and decay using concrete models, tables, graphs, or algebraic
methods.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential
Relationships; Topic Notes
Materials
TASK
Sheets of bean cutouts
Packages of Stars
One Grain of Rice by Demi
Graphing Calculators
#1
#2
#3
#4
X
X
X
X
X
X
X
Procedures
Participants should be arranged in groups of 3–4.
In this section, exponential relationships are explored similar to the manner linear
relationships in Linear Functions were explored. The connection between the
linear starting point and y-intercept is analogous to the connection between the
exponential starting point and y-intercept. The connection between the added
constant and the slope of linear functions is analogous to the constant multiplier
and the base of exponential functions. Participants are expected to write
exponential functions using a starting point and a common multiplier or ratio just
as they were expected to write linear functions using starting point and a common
difference. Encourage participants to make connections between what is
happening in the problem situation and the parameters in the exponential
functions.
Task 3.3.1:
Calculator Model for Exponential Functions
Work through Task 1 with participants.
In problem 1, participants are investigating the effect of recursive addition.
Lead participants to enter the calculator key sequence in their calculator
and observe the result:
1)
a number
Enter
operation
number;
then continuallyterter
touching Enter
2) enter as a sequence to show the count and the value.
Ask participants to enter the count and value in Table I
Ask participants to create a scatter plot with an
appropriate viewing window.
In problem 2, participants are investigating the effect of recursive multiplication.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential
Relationships; Topic Notes
Lead participants to enter the calculator key sequence in their calculator
and observe the result:
1)
a number
Enter
then continuallyterter
touching
operation
number;
Enter
2) enter as a sequence to show the count and the value.
Ask participants to enter the count and value in Table II
Ask participants to create a scatter plot with an
appropriate viewing window.
Discuss the summary questions with the
participants.
Task 3.3.2:
One Grain of Rice
Work through Task 3.3.2 with participants.
Read One Grain of Rice by Demi, stopping after Rani has been given four
grains of rice. If you do not have access to the book, you may retell the
story in your own words.
Participants determine how many grains of rice Rani received on days
1-7. Complete Table 1. Predict the number of grains of rice on the 10th
day.
Lead participants in completing the table, using language similar to the
following:
For the first day, Rani received one grain of rice.
For the second day, Rani received two grains of rice.
For the third day, Rani received four grains of rice.
After the sixth day, ask how can you write 1*2*2*2*2*2 with exponents?
(1*25)
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential
Relationships; Topic Notes
Teacher notes
When completing the table, one is more apt to operate recursively on the
previous term, continuing to multiply by 2. The emphasis here is on
expressing the number of grains of rice in terms of the number of days in
order to develop a function rule.
Ask participants to create a scatter plot with an appropriate viewing
window.
Discuss the questions with the participants.
Math notes
Recursion used in linear equations is addition. The recursion used in
exponential functions is multiplication. Repeated addition will always
create a linear model. Repeated multiplication will always create an
exponential model.
Task 3.3.3:
Stars, Stars, Stars
Work through Task 3 with participants.
Participants take a sheet of paper and fold it in half four times. Unfold the
paper. The paper is separated into 16 sections. Stars are placed on each
section of the paper according to this pattern:
• Place one star on section 1
• Place two stars on section 2
• Place four stars on section 3
(doubling the previous amount each time)
Then, participants estimate the number of stars they will need to complete
the page using this pattern. They will request the estimated number of stars
from the leader. Participants can request stars only one time.
Participants place all of their stars on the paper and complete Table 1 as
they place their stars. Lead participants in completing the table, using
language similar to the following:
In the first section, one star is placed.
In the second section, two stars are placed.
In the third section, four stars are placed.
Guide participants to write a function for the number of stars in section
number n.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential
Relationships; Topic Notes
Ask participants to create a scatter plot with an appropriate viewing
window.
Discuss questions 6 and 7 with the participants.
Task 3.3.4: Exponential Decay
Work through Task 3.3.4 with participants.
First work the problem using a smaller number of beans. Using any kind of
counters as the beans, place 87 beans in a container. Create a table and
record the total number of beans in the container. Remove one third of the
beans and place in box 1. On the table record the number of beans in box
1. Remove one-third of the beans in box 1 and place in box 2. On the table
record the number of beans in box 2. Continue removing one-third of the
beans and placing the beans in the next box until only one bean is left.
Record the number of beans in the table each time.
Ask participants to create a scatter plot with an appropriate viewing
window.
Guide participants to write a function for the number of beans in box n, if
the genie gave you 1 billion beans and the same function rule.
Reference
Demi, 1997. One Grain of Rice: A Mathematical Folktale. Scholastic Press, New
York.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.